{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:4CXAQO62KXI5PFGYM3TUSS3QT6","short_pith_number":"pith:4CXAQO62","canonical_record":{"source":{"id":"1908.06879","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2019-08-19T15:31:58Z","cross_cats_sorted":[],"title_canon_sha256":"1b0769e484bc22420e6d71f40c177abcf6912429b60323a79de9a2bdda6dc4a9","abstract_canon_sha256":"c4b90d60b93ba56b7e7c9d0f3681da52b30e3c045762a95745c29447cbafecbc"},"schema_version":"1.0"},"canonical_sha256":"e0ae083bda55d1d794d866e7494b709fbc67306f6f01b9d18579f021de48013d","source":{"kind":"arxiv","id":"1908.06879","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.06879","created_at":"2026-07-04T23:58:20Z"},{"alias_kind":"arxiv_version","alias_value":"1908.06879v1","created_at":"2026-07-04T23:58:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.06879","created_at":"2026-07-04T23:58:20Z"},{"alias_kind":"pith_short_12","alias_value":"4CXAQO62KXI5","created_at":"2026-07-04T23:58:20Z"},{"alias_kind":"pith_short_16","alias_value":"4CXAQO62KXI5PFGY","created_at":"2026-07-04T23:58:20Z"},{"alias_kind":"pith_short_8","alias_value":"4CXAQO62","created_at":"2026-07-04T23:58:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:4CXAQO62KXI5PFGYM3TUSS3QT6","target":"record","payload":{"canonical_record":{"source":{"id":"1908.06879","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2019-08-19T15:31:58Z","cross_cats_sorted":[],"title_canon_sha256":"1b0769e484bc22420e6d71f40c177abcf6912429b60323a79de9a2bdda6dc4a9","abstract_canon_sha256":"c4b90d60b93ba56b7e7c9d0f3681da52b30e3c045762a95745c29447cbafecbc"},"schema_version":"1.0"},"canonical_sha256":"e0ae083bda55d1d794d866e7494b709fbc67306f6f01b9d18579f021de48013d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T23:58:20.631272Z","signature_b64":"fVGS08d0so3SVcKorXlCRYoHG5kITy6nkCQ1xDGn3p26ZcTq2Cnwtvl3JmLZxLN7SKxaGLb6cRkgT9DH3Y04Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e0ae083bda55d1d794d866e7494b709fbc67306f6f01b9d18579f021de48013d","last_reissued_at":"2026-07-04T23:58:20.630756Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T23:58:20.630756Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1908.06879","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T23:58:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aT8Y/HkcjoeC5eWjN8YtZTLF/oixQHTdedEI4eggvD4H9WnsuE/yL43R8CKPdVdwARLAk8Kq8E1APkX8r4+VAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T04:18:42.942677Z"},"content_sha256":"9979777e8e5ca1ed54563527dfbd0848792e4f0ac6bf92a3ca8ee04d4dee1ffc","schema_version":"1.0","event_id":"sha256:9979777e8e5ca1ed54563527dfbd0848792e4f0ac6bf92a3ca8ee04d4dee1ffc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:4CXAQO62KXI5PFGYM3TUSS3QT6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Covariant & Contravariant Homotopy Theories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CT","authors_text":"Hoang Kim Nguyen","submitted_at":"2019-08-19T15:31:58Z","abstract_excerpt":"Given a locally presentable category together with a suitable functorial cylinder object, we construct model structures which are sensitive to the `direction' of the cylinder. We show that the Covariant and Contravariant model structures on simplicial sets as well as the coCartesian and Cartesian model structures on marked simplicial sets are examples of our formalism. In this setting, notions of final and initial maps and smooth and proper maps arise very naturally and we will identify these maps in the examples."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.06879","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.06879/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T23:58:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vjnfgvvPLkeaeD43xy2C2qG18wDRTp454E0HMEcYIKvaLbul5OcucHErk1EQvoPX2oSZtj2ErRYIc9HFwi7cAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T04:18:42.943268Z"},"content_sha256":"f26a7dcf6c1804213aa79ce803bd98a3b7a2d51b551cea5c719eb5e6fe22274e","schema_version":"1.0","event_id":"sha256:f26a7dcf6c1804213aa79ce803bd98a3b7a2d51b551cea5c719eb5e6fe22274e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4CXAQO62KXI5PFGYM3TUSS3QT6/bundle.json","state_url":"https://pith.science/pith/4CXAQO62KXI5PFGYM3TUSS3QT6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4CXAQO62KXI5PFGYM3TUSS3QT6/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-16T04:18:42Z","links":{"resolver":"https://pith.science/pith/4CXAQO62KXI5PFGYM3TUSS3QT6","bundle":"https://pith.science/pith/4CXAQO62KXI5PFGYM3TUSS3QT6/bundle.json","state":"https://pith.science/pith/4CXAQO62KXI5PFGYM3TUSS3QT6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4CXAQO62KXI5PFGYM3TUSS3QT6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:4CXAQO62KXI5PFGYM3TUSS3QT6","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"c4b90d60b93ba56b7e7c9d0f3681da52b30e3c045762a95745c29447cbafecbc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2019-08-19T15:31:58Z","title_canon_sha256":"1b0769e484bc22420e6d71f40c177abcf6912429b60323a79de9a2bdda6dc4a9"},"schema_version":"1.0","source":{"id":"1908.06879","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.06879","created_at":"2026-07-04T23:58:20Z"},{"alias_kind":"arxiv_version","alias_value":"1908.06879v1","created_at":"2026-07-04T23:58:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.06879","created_at":"2026-07-04T23:58:20Z"},{"alias_kind":"pith_short_12","alias_value":"4CXAQO62KXI5","created_at":"2026-07-04T23:58:20Z"},{"alias_kind":"pith_short_16","alias_value":"4CXAQO62KXI5PFGY","created_at":"2026-07-04T23:58:20Z"},{"alias_kind":"pith_short_8","alias_value":"4CXAQO62","created_at":"2026-07-04T23:58:20Z"}],"graph_snapshots":[{"event_id":"sha256:f26a7dcf6c1804213aa79ce803bd98a3b7a2d51b551cea5c719eb5e6fe22274e","target":"graph","created_at":"2026-07-04T23:58:20Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1908.06879/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Given a locally presentable category together with a suitable functorial cylinder object, we construct model structures which are sensitive to the `direction' of the cylinder. We show that the Covariant and Contravariant model structures on simplicial sets as well as the coCartesian and Cartesian model structures on marked simplicial sets are examples of our formalism. In this setting, notions of final and initial maps and smooth and proper maps arise very naturally and we will identify these maps in the examples.","authors_text":"Hoang Kim Nguyen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2019-08-19T15:31:58Z","title":"Covariant & Contravariant Homotopy Theories"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.06879","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:9979777e8e5ca1ed54563527dfbd0848792e4f0ac6bf92a3ca8ee04d4dee1ffc","target":"record","created_at":"2026-07-04T23:58:20Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"c4b90d60b93ba56b7e7c9d0f3681da52b30e3c045762a95745c29447cbafecbc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2019-08-19T15:31:58Z","title_canon_sha256":"1b0769e484bc22420e6d71f40c177abcf6912429b60323a79de9a2bdda6dc4a9"},"schema_version":"1.0","source":{"id":"1908.06879","kind":"arxiv","version":1}},"canonical_sha256":"e0ae083bda55d1d794d866e7494b709fbc67306f6f01b9d18579f021de48013d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e0ae083bda55d1d794d866e7494b709fbc67306f6f01b9d18579f021de48013d","first_computed_at":"2026-07-04T23:58:20.630756Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T23:58:20.630756Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fVGS08d0so3SVcKorXlCRYoHG5kITy6nkCQ1xDGn3p26ZcTq2Cnwtvl3JmLZxLN7SKxaGLb6cRkgT9DH3Y04Dg==","signature_status":"signed_v1","signed_at":"2026-07-04T23:58:20.631272Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.06879","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9979777e8e5ca1ed54563527dfbd0848792e4f0ac6bf92a3ca8ee04d4dee1ffc","sha256:f26a7dcf6c1804213aa79ce803bd98a3b7a2d51b551cea5c719eb5e6fe22274e"],"state_sha256":"56dc1d4489ff85f234b6906758cda22458eb385643e136a128012cc18c775d47"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"upSz74NAuuOhn2iBMwfk3rRJK1BcgOzRn4YTjlGKhE6GiTxukZ/zACTpHn9NJjUZ09vceMZPrbM40UOxXVqFAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T04:18:42.949100Z","bundle_sha256":"4c0695dde1af520d9c77f28c697439cd4066177d44c7267e3674c50ee54f5d1d"}}